Toward a universal framework for evaluating transport resistances and driving forces in membrane-based desalination processes

Desalination technologies using salt-rejecting membranes are a highly efficient tool to provide fresh water and augment existing water supplies. In recent years, numerous studies have worked to advance a variety of membrane processes with different membrane types and driving forces, but direct quantitative comparisons of these different technologies have led to confusing and contradictory conclusions in the literature. In this Review, we critically assess different membrane-based desalination technologies and provide a universal framework for comparing various driving forces and membrane types. To accomplish this, we first quantify the thermodynamic driving forces resulting from pressure, concentration, and temperature gradients. We then examine the resistances experienced by water molecules as they traverse liquid- and air-filled membranes. Last, we quantify water fluxes in each process for differing desalination scenarios. We conclude by synthesizing results from the literature and our quantitative analyses to compare desalination processes, identifying specific scenarios where each process has fundamental advantages.

where is entropy production, Jq is heat flux, T is temperature, Jw is water flux, and " is chemical potential (23,35). Gradients in chemical potential relate pressure and concentration to mass flux while a difference in temperature relates to heat flux. An equation relating the transport of both mass and heat can be derived through force-flux relationships, relating differences in chemical potential and temperature to mass and heat flux: where lij are the conductive coefficients with lqq representing Fourier conductivity, lvv correlating to the water permeability, and lvq/lqv relating to the Dufour coefficient (41). The Onsager relation lvq = lqv applies allowing for the derivation of water flux in terms of gradients in chemical potential and temperature: where Q * is the heat of transport (typically equal to a difference in partial molar enthalpy), " is the molar volume of liquid water, and A is the water permeability coefficient defined in Eq. 1 of the main text. Under the assumption of a local isothermal condition, chemical potential can be expressed in terms of pressure and concentration: where ∆ & and ∆ are the differences in hydraulic and osmotic pressure across the membrane (29). Substitution into Eq. S4 allows for water flux to be expressed in terms of pressure, concentration, and temperature: Directly comparing liquid and vapor permeability coefficients The correlation between a difference in hydraulic and osmotic pressure across a nanopore, ∆ & and ∆ , and a difference in vapor pressure, ∆ ' , was originally derived by Lee and Karnik: where ',* ( ) is the equilibrium vapor pressure of water at temperature, T (66). Equating water flux in both liquid-and air-filled membranes allows for the relation between the water permeability coefficient (A) used in Eq. 1 and the vapor permeability coefficient (Bw) used to model transport in air-filled membranes; this relation between A and Bw is shown in Eq. 8 of the main text. It is important to note that this derivation involves the use of the Kohler equation, Raoult's law for dilute solutions (<1M), and the assumption of a hydraulic pressure less than 100 bar. The relationship in Eq. S7 along with Eq. 8 from the main text can be applied to the universal water flux equation for each driving force where pressure-driven and concentration-driven processes are shown in Eq. S8.
Eq. S8 shows how with the assumption of pressure and concentration gradients as the only driving forces, Eq. 1 can be applied to both liquid-filled reverse osmosis and forward osmosis as well as air-filled pressure-driven and osmotic distillation processes. Similarly, assuming a temperature difference to be the only driving force, Eq. S9 shows how applying Eq. 8, the Clausius-Clapeyron relationship, and the Lagrange mean value theorem allows for the conversion of Eq. 1 from describing the water flux in thermo-osmosis to the water flux in membrane distillation.
Concentration polarization and temperature polarization The magnitude of concentration polarization is quantified using the concentration polarization coefficient (CPC) which is defined as the ratio of the interfacial concentration difference (∆ / ) and the bulk concentration difference (∆ 0 ), which is affected by both permeate flux and hydrodynamics of the feed stream: Film theory is usually applied to estimate the interfacial concentration. For RO, CPC is estimated as: where 3,/ , 3,0 , and 4 are feed interfacial concentration, feed bulk concentration, and permeate concentration, respectively; " is permeate flux; and 5 is the mass transfer coefficient near membrane surface and can be estimated with the Sherwood number (109). For PD, CPC is estimated as: For FO, CPC is estimated as: where 6,/ and 6,0 are draw interfacial concentration and draw bulk concentration, respectively; is the structural parameter of the support layer; is the solute diffusion coefficient; and 7 is the solute permeability (92,109). For OD, CPC is estimated as: We note that external CP on draw side is neglected for FO and OD because it is insignificant as compared to the internal CP within the support layer (109). For MD, CPC is estimated in a manner analogous to TP: The magnitude of temperature polarization is quantified using the temperature polarization coefficient (TPC), which is defined as the ratio of interfacial temperature difference (∆ / ) and bulk temperature difference (∆ 0 ) in temperature driving processes (e.g., MD and TO), which is mainly determined by water flux, membrane thermal conductivity, and bulk stream heat transfer coefficients (110,111): where " is water vapor permeability coefficient and is the density of liquid water (66,71). ' ( , , & ) is water vapor pressure as a function of temperature, concentration, and hydraulic pressure, and can be further expressed as: The equilibrium vapor pressure, ,0 ( ), can be estimated using Antoine's equation: Where A, B, and C are Antoine coefficients specific to a given substance (114). The water activity " ( ) can be estimated by an empirical equation (115): For liquid-filled membranes, volumetric water ( " ) and salt ( 7 ) fluxes can be calculated by: where and 7 are water and salt permeabilities, respectively, and are correlated by 7 = 0.0133 F ; ∆ & is the applied hydraulic pressure, Δ is thermo-osmotic pressure, and ∆ is the osmotic pressure difference (92).

Discussion of unexpectedly high mass transport rates in air-filled membranes
The current framework for vapor transport in air-filled membranes claims that thinner membranes result in a reduction in transmission resistances. However, when the air gap (or active layer) thickness is decreased, air-filled membranes suffer from an increase in temperature polarization in membrane distillation as well as more prominent interfacial resistances associated with evaporation and condensation. Thus, the maximum mass transport rate of air-filled membranes can be achieved by optimizing thickness with a theoretical upper limit set by both temperature polarization and interfacial resistances. Although most of the literature for air-filled membranes reports permeabilities within the range of the current framework, it is important to point out that a few studies have shown permeabilities beyond the theoretical upper limit (20-22, 80, 95, 116, 117). Such high fluxes may be attributable to cluster evaporation, changes in the enthalpy of vaporization, or unexpectedly high values for the condensation coefficient. Further study of these anomalously high transport rates may demonstrate that the water flux limits defined by conventional theory must be revised.
For pressure and concentration driven processes, ∆ / alone the reflects the degree of TP, as ∆ 0 is zero. Using MD as an example, interfacial temperatures can be estimated by solving the following heat transfer equations: where ! and ' are heat flux and water vapor flux, respectively; * is the heat of transport across the membrane; ℎ 9 and ℎ : are heat transfer coefficients of hot feed stream and cold permeate stream, respectively, and can be estimated with Nusselt number; and / is membrane thickness (28). The membrane thermal conductivity, / , can be estimated by the following equation: where is membrane porosity, and . andare thermal conductivity of the polymer matrix and air, respectively (28). The general approach also applies to PD, OD, and TO processes. We note that TP is usually not considered in RO and FO processes since the effect is typically negligible. For asymmetric composite membranes with a hydrophilic support layer, the heat transfer coefficient of the stream adjacent to the support layer needs to be modified to capture the internal TP effect: where 7 and 7 are thermal conductivity and thickness of the support layer, respectively (112). It is important to note that CP and TP can be related through the heat and mass analogy by the following equation: where C& is the thermal efficiency, . is the specific heat, ∆ A4 is the temperature across the temperature polarization region, ℎ 5; is the enthalpy vaporization, is the exponent of the Prandtl number in the Nusselt number correlation, and is the Lewis number that defined as the ratio of thermal diffusivity and mass diffusivity (113).

Flux calculation for air-filled and liquid-filled membranes
For air-filled membranes, volumetric water vapor flux ( " ) can be calculated by:  (25) η Probability of a gas molecule on one side of the pore reaching the other side L -pore aspect ratio (pore length/pore radius) a Adjustable constant (5) -self diffusion coefficient in water b Adjustable constant related to the size of the species Determined experimentally (5,57) , , (